In discrete-time, when the index set 𝕋 is countable, the result is as follows.

Doob’s Optional Sampling Theorem.

Suppose that the index set T is countable and that S≤T are stopping times bounded above by some constantc∈T.
If (Xt) is a martingale then XT is an integrable random variable and

𝔼[XT|ℱS]=XS,ℙ almost surely.

(1)

Similarly, if X is a submartingale then XT is integrable and

𝔼[XT|ℱS]≥XS,ℙ almost surely.

(2)

If X is a supermartingale then XT is integrable and

𝔼[XT|ℱS]≤XS,ℙ almost surely.

(3)

This theorem shows, amongst other things, that in the case of a fair casino, where your return is a martingale, betting strategies involving ‘knowing when to quit’ do not enhance your expected return.

In continuous-time, when the index set 𝕋 an interval of the real numbers, then the stopping times S,T can have a continuous distribution and XS,XT need not be measurable quantities. Then, it is necessary to place conditions on the sample paths of the process X. In particular, Doob’s optional sampling theorem holds in continuous-time if X is assumed to be right-continuous.